Theorem uspgrushgr 26541

Description: A simple pseudograph is an undirected simple hypergraph. (Contributed by AV, 19-Jan-2020.) (Revised by AV, 15-Oct-2020.)

Assertion

Ref	Expression
uspgrushgr	⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ USHGraph)

Proof of Theorem uspgrushgr

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2778	. . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		eqid 2778	. . . . 5 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
3	1, 2	isuspgr 26518	. . . 4 ⊢ (𝐺 ∈ USPGraph → (𝐺 ∈ USPGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}))
4		ssrab2 3908	. . . . 5 ⊢ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ⊆ (𝒫 (Vtx‘𝐺) ∖ {∅})
5		f1ss 6358	. . . . 5 ⊢ (((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ∧ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ⊆ (𝒫 (Vtx‘𝐺) ∖ {∅})) → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(𝒫 (Vtx‘𝐺) ∖ {∅}))
6	4, 5	mpan2 681	. . . 4 ⊢ ((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(𝒫 (Vtx‘𝐺) ∖ {∅}))
7	3, 6	syl6bi 245	. . 3 ⊢ (𝐺 ∈ USPGraph → (𝐺 ∈ USPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(𝒫 (Vtx‘𝐺) ∖ {∅})))
8	1, 2	isushgr 26426	. . 3 ⊢ (𝐺 ∈ USPGraph → (𝐺 ∈ USHGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(𝒫 (Vtx‘𝐺) ∖ {∅})))
9	7, 8	sylibrd 251	. 2 ⊢ (𝐺 ∈ USPGraph → (𝐺 ∈ USPGraph → 𝐺 ∈ USHGraph))
10	9	pm2.43i 52	1 ⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ USHGraph)

Syntax hints:   → wi 4   ∈ wcel 2107  {crab 3094   ∖ cdif 3789   ⊆ wss 3792  ∅c0 4141  𝒫 cpw 4379  {csn 4398   class class class wbr 4888  dom cdm 5357  –1-1→wf1 6134  ‘cfv 6137   ≤ cle 10414  2c2 11435  ♯chash 13441  Vtxcvtx 26361  iEdgciedg 26362  USHGraphcushgr 26422  USPGraphcuspgr 26514

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-nul 5027

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-br 4889  df-opab 4951  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fv 6145  df-ushgr 26424  df-uspgr 26516

This theorem is referenced by:  uspgrupgrushgr  26543  usgredgedg  26593  vtxdusgrfvedg  26856  1loopgrvd2  26868  isomuspgr  42761